The points (4, –5) and (– 4, 1) are endpoints of a diameter of a circle.
(a) Find the center of the circle.
(b) Find the length of the radius of the circle. (Note that this is a distance.) Give the exact answer. Show work.
(c) State the equation of the circle (in standard form). Note that you know the center and the radius from the previous parts.
average x = (4-4)/2 = 0
average y = (-5 +1)/2 = -2
so center at (0,-2)
half the distance between the two given points
d = diameter = sqrt(8^2 + 6^2) =sqrt(64+36) = sqrt(100 = 10
so radius = 5
(x-k)^2 + (y-h)^2 = r^2
here for center at (0,-2) k=0 and h = -2
x^2 + (y+2)^2 = 25
To find the center of the circle, we can use the midpoint formula. The midpoint formula states that the midpoint of a line segment with endpoints (x₁, y₁) and (x₂, y₂) is given by the coordinates:
Midpoint = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
For the given points (4, -5) and (-4, 1), the midpoint of the diameter would give us the center of the circle.
(a) Center of the circle:
Using the midpoint formula:
x-coordinate = (4 + (-4)) / 2 = 0 / 2 = 0
y-coordinate = (-5 + 1) / 2 = -4 / 2 = -2
Therefore, the center of the circle is (0, -2).
To find the length of the radius of the circle, we can use the distance formula. The distance formula is given by:
Distance = √((x₂ - x₁)² + (y₂ - y₁)²)
(b) Length of the radius:
Using the distance formula with the points (4, -5) and (0, -2) as the endpoints:
Distance = √((0 - 4)² + (-2 - (-5))²)
= √((-4)² + 3²)
= √(16 + 9)
= √25
= 5
Therefore, the length of the radius of the circle is 5 units.
To state the equation of the circle in standard form, we use the formula:
(x - h)² + (y - k)² = r²
where (h, k) represents the center of the circle, and r represents the radius.
(c) Equation of the circle:
Given the center (0, -2) and the radius of 5 units, the equation of the circle in standard form is:
(x - 0)² + (y - (-2))² = 5²
x² + (y + 2)² = 25
Therefore, the equation of the circle is x² + (y + 2)² = 25.